Exchangeable sequences driven by an absolutely continuous random measure

Abstract

Let $S$ be a Polish space and $(X_n:n\ge 1)$ an exchangeable sequence of $S$-valued random variables. Let ${\alpha }_n(\cdot )=P(X_{n+1}\in \cdot \mid X_1,\dot{.}.,X_n)$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that ${\alpha }_n\stackrel{\mathrm{weak}}{⟶}\alpha$ a.s. Two (related) problems are addressed. One is to give conditions for $\alpha \ll \lambda$ a.s., where $\lambda$ is a (nonrandom) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_1,\dot{.}.,X_n)$, $n\ge 1$, only. The other problem is to investigate whether $\Vert\alp ha_n-α\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert \cdot \Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_n)$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029-2052].

Published in at http://dx.doi.org/10.1214/12-AOP786 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Limit theorems for a class of identically distributed random variables